Whether the media is used in FCP X or FCP 7 is irrelevant.
Open one of the original (source) movie files with QuickTime. If it plays normally there, it will play (and export) normally via FCP X.
Import the original files from within FCP X by pressing Shift+Cmd+i and navigating to the original files.
Leave all the option boxes unchecked and import the footage.
Make sure the Proxy option is NOT selected (in FCP X Playback Preferences).
Select one of your clips in the browser - it will play normally.
Right-click the clip in the browser and select Open in Timeline - the clip will play normally from here.
Export the clip using Cmd+E > Current Settings.
The clip will export and playback normally.
Please note - the above is just to show you it all works OK. You should create a new Project before you start editing the clips into a movie.
Thank you for addressing my question. I actually called Apple Care and they helped me out with this problem.
Let me note that though your advice is useful, it is how I had been importing files and they were still coming out pixelated.
The solution was to use Compressor to convert the files that were DV/DVCPRO - NTSC 720 X 480 (the format saved in FCP7) to Prores 422. For some reason, the files would work fine in Quicktime, but not FCPX. Converting them to Prores 422 before importing them remedied the problem.
If anyone else has this problem, I would suggest using this technique. You can also convert to prores without compressor by right clicking on the file and click "Encode selected file" but this produced a slightly better but still pixelated Prores version.